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          <h1 class="post-title" itemprop="name headline">编程珠玑：位排序（Python实现）

            
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              <time title="创建时间：2019-09-09 20:49:48" itemprop="dateCreated datePublished" datetime="2019-09-09T20:49:48+08:00">2019-09-09</time>
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        <h1 id="问题描述"><a href="#问题描述" class="headerlink" title="问题描述"></a>问题描述</h1><ul>
<li>输入</li>
</ul><p>给出至多10,00,000个正整数的序列</p><ul>
<li>特征：</li>
</ul><p>–每个数都小于10,000,00</p><p>–数据不重复且 数据之间不存在关联关系</p><ul>
<li><p>输出：增序输出序列</p>
</li>
<li><p>约束：</p>
</li>
</ul><p>–内存容量1MB</p><p>–磁盘空间充足</p><p>–运行时间至多几分钟—最好线性时间</p><p>代码：<a href="https://github.com/XiaoZhong233/DataStructure_Python/blob/master/sort/vectorsort.py" target="_blank" rel="noopener">https://github.com/XiaoZhong233/DataStructure_Python/blob/master/sort/vectorsort.py</a></p><a id="more"></a>









<p>采用位排序可满足上面的要求。位排序的优点就是特别特别省内存空间~100万个不同的整数排序只需要32Byte的内存空间。</p>
<h1 id="什么是位排序"><a href="#什么是位排序" class="headerlink" title="什么是位排序"></a>什么是位排序</h1><p>位排序就是将n个整数转为二进制位，然后通过按顺序读出二进制达到排序的作用。</p>
<p>例如[3,7,5,2] = 00110101 然后从左往右读出该二进制串，可以获得最后的有序序列[2,3,5,7]，如果你想变成降序，那就从右往左读。</p>
<h1 id="python实现位集合"><a href="#python实现位集合" class="headerlink" title="python实现位集合"></a>python实现位集合</h1><p>借鉴了某个大佬的代码</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br><span class="line">102</span><br><span class="line">103</span><br><span class="line">104</span><br><span class="line">105</span><br><span class="line">106</span><br><span class="line">107</span><br><span class="line">108</span><br><span class="line">109</span><br><span class="line">110</span><br><span class="line">111</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># coding='utf-8'</span></span><br><span class="line"><span class="keyword">import</span> array</span><br><span class="line"></span><br><span class="line"><span class="class"><span class="keyword">class</span> <span class="title">BitSet</span><span class="params">(object)</span>:</span></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">__init__</span><span class="params">(self, capacity)</span>:</span></span><br><span class="line">		<span class="comment">#"B"类型相当于 C 语言的 unsigned char， 即占用1byte（8位），所以size大小设置为8，一个数占一个字节</span></span><br><span class="line">		self.unit_size = <span class="number">8</span></span><br><span class="line">		self.unit_count = int((capacity + self.unit_size - <span class="number">1</span>) / self.unit_size)</span><br><span class="line">		self.capacity = capacity</span><br><span class="line">		self.arr = array.array(<span class="string">"B"</span>, [<span class="number">0</span>] * self.unit_count)</span><br><span class="line">		<span class="keyword">pass</span></span><br><span class="line"></span><br><span class="line">	<span class="comment"># 是否存在为1的位</span></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">any</span><span class="params">(self)</span>:</span></span><br><span class="line">		<span class="keyword">for</span> a <span class="keyword">in</span> self.arr:</span><br><span class="line">			<span class="keyword">if</span> a != <span class="number">0</span>:</span><br><span class="line">				<span class="keyword">return</span> <span class="literal">True</span></span><br><span class="line">		<span class="keyword">return</span> <span class="literal">False</span></span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">all</span><span class="params">(self)</span>:</span></span><br><span class="line">		<span class="comment">#是否所有位都为 1， 即是否存在置为 0 的位</span></span><br><span class="line">		t = (<span class="number">1</span> &lt;&lt; self.unit_size) - <span class="number">1</span></span><br><span class="line">		<span class="keyword">for</span> a <span class="keyword">in</span> self.arr:</span><br><span class="line">			<span class="keyword">if</span> (a &amp; t) != t:</span><br><span class="line">				<span class="keyword">return</span> <span class="literal">False</span></span><br><span class="line">		<span class="keyword">return</span> <span class="literal">True</span></span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">none</span><span class="params">(self)</span>:</span></span><br><span class="line">		<span class="comment">#是否所有位都为 0，即是否不存在置为 1 的位</span></span><br><span class="line">		<span class="keyword">for</span> a <span class="keyword">in</span> self.arr:</span><br><span class="line">			<span class="keyword">if</span> a != <span class="number">0</span>:</span><br><span class="line">				<span class="keyword">return</span> <span class="literal">False</span></span><br><span class="line">			<span class="keyword">return</span> <span class="literal">True</span></span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">count</span><span class="params">(self)</span>:</span></span><br><span class="line">		<span class="comment"># 1 的位的个数</span></span><br><span class="line">		c = <span class="number">0</span></span><br><span class="line">		<span class="keyword">for</span> a <span class="keyword">in</span> self.arr:</span><br><span class="line">			<span class="keyword">while</span> a &gt; <span class="number">0</span>:</span><br><span class="line">				<span class="keyword">if</span> a &amp; <span class="number">1</span>:</span><br><span class="line">					c+=<span class="number">1</span></span><br><span class="line">				a = a&gt;&gt;<span class="number">1</span></span><br><span class="line">				<span class="keyword">pass</span></span><br><span class="line">		<span class="keyword">return</span> c</span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">size</span><span class="params">(self)</span>:</span></span><br><span class="line">		<span class="comment">#所有位的个数</span></span><br><span class="line">		<span class="keyword">return</span> self.unit_count * self.unit_size</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">get</span><span class="params">(self, pos)</span>:</span></span><br><span class="line">		<span class="comment">#获取第 pos 位的值</span></span><br><span class="line">		index = int(pos / self.unit_size)</span><br><span class="line">		offset = (self.unit_size - (pos - index * self.unit_size) - <span class="number">1</span>) % self.unit_size</span><br><span class="line">		<span class="keyword">return</span> (self.arr[index] &gt;&gt; offset) &amp; <span class="number">1</span>	</span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">test</span><span class="params">(self, pos)</span>:</span></span><br><span class="line">		<span class="comment">#判断第 pos 位的值是否为 1</span></span><br><span class="line">		<span class="keyword">if</span> self.get(pos):</span><br><span class="line">			<span class="keyword">return</span> <span class="literal">True</span></span><br><span class="line">		<span class="keyword">return</span> <span class="literal">False</span></span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">set</span><span class="params">(self, pos = <span class="number">-1</span>)</span>:</span></span><br><span class="line">		<span class="comment">#设置第 pos 位的值为 1，若 pos 为 -1， 则所有位都置为 1</span></span><br><span class="line">		<span class="keyword">if</span> pos &gt;= <span class="number">0</span>:</span><br><span class="line">			index = int(pos / self.unit_size)</span><br><span class="line">			offset = (self.unit_size - (pos - index * self.unit_size) - <span class="number">1</span>) % self.unit_size</span><br><span class="line">			self.arr[index] = (self.arr[index]) | (<span class="number">1</span> &lt;&lt; offset)</span><br><span class="line">		<span class="keyword">else</span>:</span><br><span class="line">			t = (<span class="number">1</span> &lt;&lt; self.unit_size) - <span class="number">1</span></span><br><span class="line">			<span class="keyword">for</span> i <span class="keyword">in</span> range(self.unit_count):</span><br><span class="line">				self.arr[i] = self.arr[i] | t</span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">reset</span><span class="params">(self, pos = <span class="number">-1</span>)</span>:</span></span><br><span class="line">		<span class="comment">#设置第 pos 位的值为 0，若 pos 为 -1， 则所有位都置为 0</span></span><br><span class="line">		<span class="keyword">if</span> pos &gt;= <span class="number">0</span>:</span><br><span class="line">			index = int(pos / self.unit_size)</span><br><span class="line">			offset = (self.unit_size - (pos - index * self.unit_size) - <span class="number">1</span>) % self.unit_size</span><br><span class="line">			x = (<span class="number">1</span> &lt;&lt; offset)</span><br><span class="line">			self.arr[index] = (self.arr[index]) &amp; (~x)</span><br><span class="line">		<span class="keyword">else</span>:</span><br><span class="line">			<span class="keyword">for</span> i <span class="keyword">in</span> range(self.unit_count):</span><br><span class="line">				self.arr[i] = <span class="number">0</span></span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">flip</span><span class="params">(self, pos = <span class="number">-1</span>)</span>:</span></span><br><span class="line">		<span class="comment">#把第 pos 位的值取反，若 pos 为 -1， 则所有位都取反</span></span><br><span class="line">		<span class="keyword">if</span> pos &gt;= <span class="number">0</span>:</span><br><span class="line">			<span class="keyword">if</span> self.get(pos):</span><br><span class="line">				self.reset(pos)</span><br><span class="line">			<span class="keyword">else</span>:</span><br><span class="line">				self.set(pos)</span><br><span class="line">		<span class="keyword">else</span>:</span><br><span class="line">			<span class="keyword">for</span> i <span class="keyword">in</span> range(self.unit_count):</span><br><span class="line">				self.arr[i] = ~self.arr[i] + (<span class="number">1</span> &lt;&lt; self.unit_size)</span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">binstr</span><span class="params">(self)</span>:</span></span><br><span class="line">		b = <span class="string">''</span></span><br><span class="line">		<span class="keyword">for</span> a <span class="keyword">in</span> self.arr:</span><br><span class="line">			t = bin(a)</span><br><span class="line">			b += <span class="string">"0"</span> * (self.unit_size - len(t) + <span class="number">2</span>) + t + <span class="string">","</span></span><br><span class="line">		<span class="keyword">return</span> <span class="string">"["</span> + b.replace(<span class="string">"0b"</span>, <span class="string">""</span>).strip(<span class="string">","</span>) + <span class="string">"]"</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">show</span><span class="params">(self)</span>:</span></span><br><span class="line">		<span class="keyword">return</span> self.arr</span><br><span class="line"></span><br><span class="line"><span class="keyword">if</span> __name__ == <span class="string">'__main__'</span>:</span><br><span class="line">	bitSet = BitSet(<span class="number">32</span>)</span><br><span class="line">	bitSet.set()</span><br><span class="line">	print(bitSet.binstr())</span><br><span class="line">	print(bitSet.count())</span><br></pre></td></tr></table></figure>

<h1 id="位排序（Python实现）"><a href="#位排序（Python实现）" class="headerlink" title="位排序（Python实现）"></a>位排序（Python实现）</h1><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># coding = "utf-8"</span></span><br><span class="line"><span class="keyword">import</span> sys</span><br><span class="line"></span><br><span class="line">sys.path.append(<span class="string">'../tool'</span>)</span><br><span class="line"><span class="keyword">import</span> TimeCalculator</span><br><span class="line"><span class="keyword">import</span> BitSet</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">generate_data</span><span class="params">(n)</span>:</span></span><br><span class="line">    <span class="keyword">return</span> np.random.randint(low=<span class="number">0</span>, high=n, size=n)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 写成txt文件</span></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">wirte_data</span><span class="params">(n)</span>:</span></span><br><span class="line">    data = generate_data(n)</span><br><span class="line">    np.savetxt(<span class="string">'./data.txt'</span>, X=data, fmt=<span class="string">"%d"</span>,delimiter=<span class="string">" "</span>, newline = <span class="string">" "</span>)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 写成二进制文件</span></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">wirte_data2</span><span class="params">(n)</span>:</span></span><br><span class="line">    data = generate_data(n)</span><br><span class="line">    np.save(file =<span class="string">'./data'</span>, arr = data, )</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">load_data2</span><span class="params">(file = <span class="string">'./data.npy'</span>)</span>:</span></span><br><span class="line">    <span class="keyword">return</span> np.load(file)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="comment"># 对n（n&lt;=1000000）个小于n的正整数序列进行排序</span></span><br><span class="line"><span class="comment"># 时间在10s以内</span></span><br><span class="line"><span class="comment"># 空间占用小于1MB</span></span><br><span class="line"><span class="meta">@TimeCalculator.display_time</span></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">vertorsort</span><span class="params">(data, n=<span class="number">1000000</span>)</span>:</span></span><br><span class="line">    bitset = BitSet.BitSet(n)</span><br><span class="line">    <span class="comment"># 初始化位图,模拟从I/O读入数据</span></span><br><span class="line">    <span class="keyword">for</span> i <span class="keyword">in</span> data:</span><br><span class="line">        bitset.set(i)</span><br><span class="line">    result = []</span><br><span class="line">    <span class="comment"># 输出位图排序结果,模拟向I/0写入数据</span></span><br><span class="line">    print(<span class="string">"空间大小占用:"</span>)</span><br><span class="line">    print(str(sys.getsizeof(bitset)) + <span class="string">"Byte"</span>)</span><br><span class="line">    <span class="keyword">for</span> i <span class="keyword">in</span> range(bitset.size()):</span><br><span class="line">        <span class="keyword">if</span> bitset.test(i):</span><br><span class="line">            result.append(i)</span><br><span class="line">    <span class="keyword">return</span> result</span><br><span class="line">    <span class="keyword">pass</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="keyword">if</span> __name__ == <span class="string">'__main__'</span>:</span><br><span class="line">    <span class="comment"># 模拟100万个100万内的数据</span></span><br><span class="line">    wirte_data2(<span class="number">10</span>**<span class="number">6</span>)</span><br><span class="line">    <span class="comment"># 模拟从磁盘中读取数据</span></span><br><span class="line">    data = load_data2()</span><br><span class="line">    <span class="comment"># 模拟向I/O输出排序结果</span></span><br><span class="line">    result = vertorsort(data, <span class="number">10</span>**<span class="number">6</span>)</span><br><span class="line">    np.savetxt(<span class="string">'./datasort.txt'</span>, X=result, fmt=<span class="string">"%d"</span>,delimiter=<span class="string">" "</span>, newline = <span class="string">" "</span>)</span><br><span class="line">    <span class="comment"># print(result)</span></span><br></pre></td></tr></table></figure>

<h1 id="运行结果"><a href="#运行结果" class="headerlink" title="运行结果"></a>运行结果</h1><h2 id="时间与空间花费"><a href="#时间与空间花费" class="headerlink" title="时间与空间花费"></a>时间与空间花费</h2><p><img src="https://zhong-blog.oss-cn-shenzhen.aliyuncs.com/blog/20190910143008.png!blog" alt="20190910143008"></p>
<h2 id="输出结果"><a href="#输出结果" class="headerlink" title="输出结果"></a>输出结果</h2><p><img src="https://zhong-blog.oss-cn-shenzhen.aliyuncs.com/blog/20190910143135.png!blog" alt="20190910143135"></p>
<h1 id="优点与缺点"><a href="#优点与缺点" class="headerlink" title="优点与缺点"></a>优点与缺点</h1><ul>
<li>优点：</li>
</ul>
<p>位排序非常非常非常节省内存空间，100万个不同的整数排序，只需要32B的辅助空间。</p>
<ul>
<li>缺点</li>
</ul>
<p>数据需要从I/O读取是硬伤，实际上正是由于了I/O拖累了整体的速度，100万个整数排序，需要4600ms（4.6秒）左右（如果采用C语言应该会更快），时间成本上比快速排序差远了。</p>
<p>如果使用快速排序，100万个整数差不多需要4~5MB的内存空间，非常大了，是位排序的15万倍。但快排的时间成本也是非常可观的，只需要400ms（0.4秒）作用。</p>
<p>下图是非递归快排的100万不同随机整数的排序时间花费：</p>
<p><img src="https://zhong-blog.oss-cn-shenzhen.aliyuncs.com/blog/20190910144020.png!blog" alt="20190910144020"></p>
<p>快速排序果然是内部排序之王</p>
<h1 id="适用场景"><a href="#适用场景" class="headerlink" title="适用场景"></a>适用场景</h1><p>位排序的适用场景是内存空间十分有限，但速度没那么严格的情况，典型的时间换空间。</p>

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